Hence, the path-addition
operation distinguishes the graph classes RAC,
FCF, \(k\)GAP (\(k \ge 1\) ), and QUASI
from FAN,
1PLANAR and 2PLANAR,
respectively. Since every 1-planar graph is fan-crossing free
[1]}, every 2-planar graph is 1-gap-planar
[2]}, and every fan-crossing or 2-planar graph [3]}
is quasi-planar,
it provides an alternative proof for the
properness of these inclusions.
